It took almost 30 years to see that such an identity (REF ) is not accidental but
also exists for a general group \(\Gamma _0(N)\)  (see [1]}).
Now it seems natural to ask if one can get such a relation for general automorphic forms.
In this paper we attempt to get such a formula, namely, an identity between
a generating function of periods of Hilbert modular forms over totally real number fields with strict class number one and Jacobi forms (see Theorem REF ). The function
\(F_{\tau }(u,v) \)  in (REF ) was introduced by Kronecker in a more general form (see page 70 in [2]}) and several properties of that have been explored by Zagier [3]}. The essential property of \(F_{\tau }(u,v) \)  is that it
can be identified as a sum of derivatives of Eisenstein series (called the "Kuznetsov lifting") and using this fact we are able to extend Zaiger' identity to that for a totally real number field. The main result of this paper shows the first connection between the Kronecker series and the critical \(L\) - values of Hilbert modular forms over the totally real number fields. It also gives a systematic way to compute Hilbert Hecke eigenforms and the special values of \(L\)  -functions by taking the expansions of the Kronecker series.
